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arXiv · 2407.12398

Polynomial convergence rate at infinity for the cusp winding spectrum of generalized Schottky groups

Yuya Arima
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Abstract

We show that the convergence rate of the cusp winding spectrum to the Hausdorff dimension of the limit set of a generalized Schottky group with one parabolic generator is polynomial. Our main theorem provides the new phenomenon in which differences in the Hausdorff dimension of the limit set generated by a Markov system cause essentially different results on multifractal analysis. This paper also provides a new characterization of the geodesic flow on the Poinca\'re disc model of two-dimensional hyperbolic space and the limit set of a generalized Schottky group. To prove our main theorem we use thermodynamic formalism on a countable Markov shift, gamma function, and zeta function.

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Yuya Arima. 2024-07-17. Polynomial convergence rate at infinity for the cusp winding spectrum of generalized Schottky groups. https://arxiv.org/abs/2407.12398

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